Real and Rational Numbers

Date: 02/27/2001 at 14:02:38
From: Eileen Bach
Subject: Real and rational numbers
How can I show that the number of rational numbers between 0 and 1 is
the same as the number of natural numbers? (considering the following
ordering of these fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)

Date: 02/28/2001 at 14:17:57
From: Doctor Floor
Subject: Re: Real and rational numbers
Hi, Eileen,
Thanks for writing.
By having the ordering as you present it, you know you can count the
rationals between 0 and 1. But counting here is the same as making a
function
f: (0,1)/\Q ---> N
[ (0,1)/\Q is the open interval from 0 to 1 of the rational numbers ]
which is to say, a function giving for each rational number between 0
and 1 a natural number.
By this function each natural number is reached exactly once. And of
course each rational between 0 and 1 is mapped to a natural number.
Such a function we call a 'bijection'.
If two sets are mapped to each other by a bijection, then their number
of elements is equal.
If you have more questions, just write back.
Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/